Stericated 7-orthoplexes

Orthogonal projections in B₆ Coxeter plane
7-orthoplex	Stericated 7-orthoplex	Steritruncated 7-orthoplex
Bisteritruncated 7-orthoplex	Stericantellated 7-orthoplex	Stericantitruncated 7-orthoplex
Bistericantitruncated 7-orthoplex	Steriruncinated 7-orthoplex	Steriruncitruncated 7-orthoplex
Steriruncicantellated 7-orthoplex	Steriruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex.

There are 24 unique sterication for the 7-orthoplex with permutations of truncations, cantellations, and runcinations. 14 are more simply constructed from the 7-cube. This polytope is one of 127 uniform 7-polytopes with B₇ symmetry.

Stericated 7-orthoplex

Stericated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Small cellated hecatonicosaoctaexon (acronym: scaz) (Jonathan Bowers)^[1]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steritruncated 7-orthoplex

Steritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Cellitruncated hecatonicosaoctaexon (acronym: catz) (Jonathan Bowers)^[2]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Bisteritruncated 7-orthoplex

Bisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,5{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Bicellitruncated hecatonicosaoctaexon (acronym: boctaz) (Jonathan Bowers)^[3]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Stericantellated 7-orthoplex

Stericantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Cellirhombated hecatonicosaoctaexon (acronym: craze) (Jonathan Bowers)^[4]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Stericantitruncated 7-orthoplex

Stericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Celligreatorhombated hecatonicosaoctaexon (acronym: cogarz) (Jonathan Bowers)^[5]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Bistericantitruncated 7-orthoplex

Bistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,3,5{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Bicelligreatorhombated hecatonicosaoctaexon (acronym: becogarz) (Jonathan Bowers)^[6]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncinated 7-orthoplex

Steriruncinated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Celliprismated hecatonicosaoctaexon (acronym: copaz) (Jonathan Bowers)^[7]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncitruncated 7-orthoplex

Steriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Celliprismatotruncated hecatonicosaoctaexon (acronym: captaz) (Jonathan Bowers)^[8]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncicantellated 7-orthoplex

Steriruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Celliprismatorhombated hecatonicosaoctaexon (acronym: caparz) (Jonathan Bowers)^[9]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncicantitruncated 7-orthoplex

Steriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4{3⁵,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₇, [4,3⁵]
Properties	convex

Alternate names

Great cellated hecatonicosaoctaexon (acronym: gocaz) (Jonathan Bowers)^[10]

Images

Orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Notes

^ Klitzing, (x3o3o3o3x3o4o - scaz)
^ Klitzing, (x3x3o3o3x3o4o - catz)
^ Klitzing, (o3x3x3o3o3x4o - boctaz)
^ Klitzing, (x3o3x3o3x3o4o - craze)
^ Klitzing, (x3x3x3o3x3o4o - cogarz)
^ Klitzing, (o3x3x3x3o3x4o - becogarz)
^ Klitzing, (x3o3o3x3x3o4o - copaz)
^ Klitzing, (x3x3o3x3x3o4o - captaz)
^ Klitzing, (x3o3x3x3x3o4o - caparz)
^ Klitzing, (x3x3x3x3x3o4o - gocaz)

References

H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
  - (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". x3o3o3o3x3o4o - scaz, x3x3o3o3x3o4o - catz, o3x3x3o3o3x4o - boctaz, x3o3x3o3x3o4o - craze, x3x3x3o3x3o4o - cogarz, o3x3x3x3o3x4o - becogarz, x3o3o3x3x3o4o - copaz, x3x3o3x3x3o4o - captaz, x3o3x3x3x3o4o - caparz, x3x3x3x3x3o4o - gocaz

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[1] Klitzing, (x3o3o3o3x3o4o - scaz)

[2] Klitzing, (x3x3o3o3x3o4o - catz)

[3] Klitzing, (o3x3x3o3o3x4o - boctaz)

[4] Klitzing, (x3o3x3o3x3o4o - craze)

[5] Klitzing, (x3x3x3o3x3o4o - cogarz)

[6] Klitzing, (o3x3x3x3o3x4o - becogarz)

[7] Klitzing, (x3o3o3x3x3o4o - copaz)

[8] Klitzing, (x3x3o3x3x3o4o - captaz)

[9] Klitzing, (x3o3x3x3x3o4o - caparz)

[10] Klitzing, (x3x3x3x3x3o4o - gocaz)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]